J . Phys. Chem. 1984, 88, 4823-4829 The dynamical-state representation enables us to apply the methods, since the transitions are made to occur locally at avoided crossings of the dynamical potential energy hypersurfaces. The kinetic energy operator T of a triatomic system can be written as follows in terms of the Jacobi coordinates (see Figure 17):45 1 T = -PR2 2r
+
1 -P;+ 2m
-+ ?j 2pR2
2pr2
(7.4)
i a r ar
(7.5)
a ae
pe = -ih-
= mA(mB + mC)/(mA + mB
+ mC)
H = _h2_1 -p3N-4 a 2r p 3 ~ - 4 ap
a
-+H
ap
dyn
(7.9) (7.10)
i a PR = -ih- -R R dR
/l
The hyperradius p plays the role of internuclear distance R . The total Hamiltonian in the body-fixed frame can be generally expressed as46
(7.1)
where
P, = -ih- -r
4823
(7.7)
where N reRresents the number of particles in the system, A(Q) represents the grand angular momentum, and Q represents the totality of the hyperspherical angle variables and Euler angles. of eq The dynamical states are defined as the eigenstates of Hdyn 7.10, Le., the eigenstates of the rotating complex at fixed hyperradius p, and are dependent parametrically on p. The spectrum of eigenvalues thus obtained is discrete at all p . The problem can thus be reduced to a one-dimensional problem for which we have good mathematical tools. Transitions among the dynamical states occur at avoided crossings. Namely, what are mentioned in section 4 hold basically true to any transition in any process. There arise, of course, complications mainly because the number of internal states is enormous in the case of a complicated system. The avoided crossings would appear frequently in groups along the potential ridges and thus could not be considered isolated. A new analytic theory should be developed to tackle this problem.47 A further ambitious idea is to count even electrons as the independent constituent particles to define the hyperradius p so that one could treat uniformly both electron correlation and nonadiabatic couplings, There has been considerable progress in understanding the electron correlation in two-electron excited states of atoms by using the hyperspherical The situation there is quite similar to that here. Namely, the adiabatic potential energies obtained as a function of the hyperradius of two electrons avoid crossings with each other. It should be possible to combine the two worlds extended in the different degrees of freedom by using hyperspherical coordinates of the total system composed of electrons and nuclei.42 I
m = mBmC/(mB + mC)
(7.8)
m, is the mass of atom a,J is the total angular momentum, and j is the electronic angular momentum of the system. The dynamical states are defined as the eigenstates of the electronic Hamiltonian plus the operators in eq 7.1, which do not include PR, P,, and PO, and are parametrically dependent on R , r, and 6. The last term in eq 7.3 is somewhat disturbing, because this involves J, -j,. This term is expected, however, to contribute only in the very rare case that three potential surfaces come close together. One defect of this DS representation approach is that one has to use the Jacobi coordinates (R,r,O)even to run simple classical trajectories. The dynamical states are interpreted as the eigenstates of the rotating collision complex at its fixed size. The basic idea of using these states as bases can, in principle, be generalized to more complicated systems by employing the hyperspherical coordinates.
Acknowledgment. I am indebted to Dr. H. Takagi, M. Namiki, and R. Hirokawa for their helpful discussions. I would also like to thank R. Hirokawa for her computational assistance. Numerical calculations have been carried out at the computer center of the Institute for Molecular Science.
(45) See, for instance, F. Rebentrost, “Theoretical Chemistry, Theory of Scattering: Papers in Honor of Henry Eyring”, Vol. 6, Part B, D. Henderson, Ed., Academic Press, New York, 1981, pp 3-77.
(46) J. L. Blatt and M. Fabre de la Ripelle, Ann. Phys. ( N . Y . ) ,127, 62 (1980). (47) U. Fano, Phys. Rev. A , 22, 2660 (1980). (48) U. Fano, Rep. Prog. Phys., 46, 97 (1983).
Remarks on “Quantum Chaos” Philip Pechukas Department of Chemistry, Columbia University, New York, New York 10027 (Received: November 28, 1983)
Three aspects of “quantum chaos” are discussed: ergodicity, loss of memory of the initial state, and the distribution of eigenvalues to be expected in the spectrum of a “chaotic” system.
1. Introduction
If Newton had been right and the world ran according to the laws of classical mechanics, the internal motions of a vibrating polyatomic would typically be “quasi-periodic” or “regular”’ at
low energies of excitation and “chaotic” or “irregular”’ at higher energies2 Quasi-periodic motion is characterized by a number of independent constants of the motion equal to the number of degrees of freedom of the system; “chaotic” motion is a catchall
(1) The terms “regular”and “irregular”,which serve as useful adjectives in both classical and quantum mechanics, were introduced by Percival; see I. C. Percival, J . Phys. B, 6, L229 (1973); Ado. Chem. Phys., 36, 1 (1977).
(2) For an excellent review of work on both classical and quantum aspects of molecular motion, see D. W. floid, M. L. Koszykowski, and R. A. Marcus, Annu. Rev. Phys. Chem., 32, 267 (1981).
0022-3654/84/2088-4823$01 S O / O
0 1984 American Chemical Society
4824
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984
term for more erratic behavior, of which the mildest interesting form is ergodicity, characterized by only one constant of the motion, the energy. The question whether there is an analogous distinction in quantum dynamics between “regular” and ”irregular” motion-Is there “quantum chaos”?-has been debated almost since the invention of quantum theory. The trouble with the question, and the reason discussion continues, is that all three possible answers-yes, no, and maybe-are perfectly defensible. I personally take the bold and uncompromising position that the correct answer is “maybe”; I believe that well-defined classical adjectives such as “ergodic” and “mixing” have no quantum counterparts that are both mathematically precise and physically useful, but I also believe that quantum systems that are sufficiently “near classical” may siknal the underlying classical chaos by characteristic patterns in the dpectral distribution of their eigenvalues and the spatial distribution of their eigenfunctions. However, before considering these patterns (see section IV) it is useful to examine briefly the two more definite answers to the question of quantum chaos. 1. No, there is obviously no such thing as quantum chaos: quantum dynamics is always ”regular” and is just like quasi-periodic motion in classical mechanics. To see this consider quantum motion in an N-dimensional ”energy shell” spanned by the eigenfunctions 41, ..., 4N belonging to N consecutive energy eigenvalues E l , ..., EN. Every state is specified by N complex expansion coefficients a l , ..., aNand evolves as $(t) = Xa,(t)$, where an(t) = a,(O) exp(-iE,t/h). The “phase space” in which this motion takes place is the real 2N-dimensional space in which the “coordinates” and “momenta” are respectively the real and imaginary parts of the N expansion coefficients. In this 2N-dimensional phase space we have N constants of the motion, lall -- cl, ... la#l = c;, motion takes place on the N-dimensional invariant torus (a,= cj exp(-i6’J); the equations of motion of the “angle” variables 0, are 0, = ujwhere the frequencies are uf = E,/h. Quantum motion in an N-dimensional energy shell is just classical quasi-periodic motion of a system with N degrees of freedom. The argument of the preceding paragraph is unassailable; it is also independent of h, and therefore denies that classical chaos can “emerge” from the quantum mechanics as h 0. Rather, chaos exists only at h = 0, in classjcal mechanic^;^ which is certainly an answer to the question, but perhaps not totally satisfying, since h # 0 in the world around us and yet chaos abounds. 2. Yes, there are obviously ergodic quantum systems. In fact, any system whose energy spectrum is nondegenerate must be ergodic, for then energy is the only constant of the motion, in the sense that any operator that commutes with the Hamiltonian can be expressed as a function of the Hamiltonian, by the following familiar construction: suppose H+, = E,#, where all E, are nondegenerate; if [ A , a = 0, each 4, must also be an eigenfunction of A , A d f = oldr;then A = f ( H ) wheref(x) is any function of a single real variable that assumes the values a, at n = E,. The d a i n objection to this notion of quantum ergodicity is well-known: too many systems qualify. For example, if H is a hz where h, and hz act on separable Hamiltonian, H = hl functions of different variables, the spectrum of H may be nondegenerate even though there are two constants of the motion, hl and ha- In this case the mathematician will insist that hl and hz are not independent, since both are functions of H, but one should ignore him. In the famous paper than began quantum ergodic theory: von Neumdnn considered motion within an N-dimensional energy shell and added a second spectral condition to the requirement that all N energy eigenvalues be different; the additional condition is that all paifwise differences of energy eigenvalues be different. With these two conditions von Neumann established an inequality
-
+
(3) This point can be made in various ways; for a particularly interesting way, see R. Kosloff and S . A. Rice, J. Chem. Phys., 74, 1340 (1981); see also P. Pechukas, J . Phys. Chem., 86, 2239 (1982). (4) J. von Neumann, Z. Phys., 57, 30 (1929).
Pechukas that he thought gave a dynamical foundation to quantum erogodic theory. The story is interesting and has a recent twist, so we begin in section I1 with the main inequality of quantum ergodic theory.
11. An Inequality in Quantum Ergodic Theory Von Neumann4 proposed the following calculation: let S be an N-dimensional energy shell and $(t) a normalized state evolving entirely within S; let P be the projector onto an M-dimensional subspace of S; calculate the mean square deviation of the “occupation probability” P(t) = ($(t),P$(t)) from the statistical expectation f = M / N , where the mean is a time average: T-lim
T1s,ht
(P(t)
-A2
(2.1)
The result of course depends on the projector P as well as on the state $(t). For mathematical convenience, von Neumann averaged over all M-dimensional projectors, an operation that can be rationalized on the grounds that there is no fundamental reason to prefer one P to another. Using the two spectral conditions of nondegeneracy of energy eigenvalues and of their pairwise differences, von Neumann then derived an inequality that can be written
((P-AZ)p/f2 < 2(1 - A / M
(2.2)
where ( ) p denotes the average over P‘s. If M is large, the average mean dispersion in the occupation probability is small compared to the average occupation probability$ The physical interpretation of von Neumann’s result is that a typical “macro-observer”-a typical projector P of large dimension-will rarely find substantial deviation from the statistical expectation f i n a system evolving under von Neumann’s two spectral conditions. There matters stood until the 195O’s, when the von Neumann result was repeatedly shelled; by the end of the decade there was nothing left of it. First FierzS and ter Haar6 showed that the second spectral condition, on energy differences, is unnecessary; then Farquhar and Landsberg’ showed that the assumption of eigenvalue nondegeneracy is likewise unnecessary; finally Bocchieri and Loingers showed that time-averaging-or, indeed, any dynamics whatever-is also unnecessary. Inequality (2.2) will still hold if the Hamiltonian is identically equal to zero in the energy shell S and all states are completely time independent: von Neumann’s inequality is entirely a consequence of the averaging over “macro-observers” and has nothing to do with quantum dynamics. The work of Bocchieri and Loinger is notable for its mathematical elegance and also for two aspects that I find somewhat ironic. The first irony is that in the course of demonstrating that von Neumann’s inequality is without physical content Bocchieri and Loinger substantially strengthened its mathematical content. They showed that for any normalized $ in S the average over P can be calculated exactly (($,P$)z)P
= M ( M + 1)/N(N + 1)
(2.3)
which implies inequality (2.2) in the sharper form
The second irony has to do with a subsequent paper9 in which Bocchieri and Loinger, objecting to von Neumann’s averaging over ”macro-observers”, proposed instead an averaging over all initial states belonging to S , which averaging we denote by ( )+. Of course, as they pointed out, there is little mathematical difference between averaging over P for fixed $ and averaging over $ for (5) M. Fierz, Helu. Phys. Acta, 28, 705 (1955). (6) D. ter Haar, Reu. Mod. Phys., 27, 289 (1955). (7) I. E. Farquhar and P. T. Landsberg, Proc. R . SOC.London, Ser. A , 239, 134 (1957). (8) P. Bocchieri and A. Loinger, Phys. Reu., 111, 668 (1958). (9) P. Bocchieri and A. Loinger, Phys. Reu., 114, 948 (1959).
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4825
Remarks on Quantum Chaos fixed P; it is a question of interpretation. We now have, for any
P
+ 1 ) / N ( N + 1) (2.5) but the interpretation is that, given P with M >> 1, for “most” (($J’$)’)+
= M(M
$ the probability ($,P$) does not differ much from the statistical expectation f = M/N. Inequality (2.4) still holds, in the form
((P)$/f < (1 - A / M
(2.6)
and is just as devoid of physical content. The irony is that had Bocchieri and Loinger been less concerned with the defects of the von Neumann approach-had they in fact combined their average over $ with von Neumann’s average over P-they would have discovered a much sharper form of the basic inequality, a form which in fact has some physical content in that it discriminates between time evolution according to Hamiltonians with different degeneracy structures. This is the recent twist in the story,1° and it requires a few words of introduction. From classical ergodic theory we know that the time average of real interest is not that studied by von Neuman, eq 2.1, but rather the time-averaged occupation probability itself T
P , = lim T 1 x dt P(t) T--
< (1 - A / M ((P+-A2)+/f < (1 - A / M
(2.8a) (2.8b)
We do much better by calculating the double average ( ( P $ ) + ) p One finds1° that if E , are the distinct eigenvalues of H in S and n, their degeneracies, then
( ( ( P +-A2),)p=f(l -fl(Zn,2 - l ) / ( N - 1)(N + 1)’ (2.9) a and from this we get the final, “sharp”, form of the basic inequality in quantum ergodic theory
Notice that this reduces to (2.4) when the energy shell S is completely degenerate, n, = N, while the dispersion in P+ is least when the spectrum of H in S is completely nondegenerate, n, = 1 for all a. The interpretation of eq 2.9 or 2.10 is that if one selects a state “at random” from an energy shell and determines, as a function of time, the probability that it lies in a “typical” subspace of the shell, the time average of this probability is liable to be much closer to the statistical expectation than is its instantaneous value, the more so the less degenerate the spectrum of the Hamiltonian. Is this the last word in quantum ergodic theory? One hopes not. These calculations of infinite-time averages, guided by analogy to classical ergodic theory, are rather dull stuff. In classical mechanics time averaging over an infinite interval is essential to distinguish between ergodic and nonergodic motion; in quantum mechanics time averaging over an infinite interval destroys all dynamical distinctions except those associated with eigenvalue degeneracy. Although it is nice to know that the von Neumann-Bocchieri-Loinger approach to quantum ergodic theory, as expressed in eq 2.9, does provide some dynamical justification for the natural prejudice that nondegeneracy of the energy spectrum is the minimal requirement for “quantum chaos”, it is the behavior of quantum systems over finite time intervals that is of real interest, because “quantum chaos” must have to (10) P. Pechukas, J . Math. Phys., 25, 532 (1984).
111. “Survival Probabilities” and Quantum Ergodicity
The subject of this section is a somewhat different time-dependent occupation probability, the so-called “survival probability” P(t) = l($(0),$(t))12that a system, evolving with the normalized wave function $ ( t ) , is still in the initial state $(O). That there should be a connection between survival probabilities and statistical behavior is suggested by classical ergodic theory. Consider a constant energy surface f2 in phase space and let p be the Liouville measure on this surface, normalized so that p(f2) = 1. If S is a subset of this surface and S, is its image after time t
is the “survival probability” of the classical “state” S,, as determined by its overlap with the initial set S. Classical ergodic theory is concerned with the infinite-time average of P ( t )
(2.7)
How close is P+ to the statistical expectation f! The answer depends on both the initial state $ and the projector P. Bocchieri and Loingerg pointed out that since the square of an average - is never greater than the average of the square, P$ IP(t)2 and therefore the dispersion in Ppis subject to the same uninformative inequalities as above
( ( P +-A”,p/f
do with the pattern of the energy spectrum-with the distribution of spacings between energy levels, not just with the question of how many of these spacings happen to be zero. In the next section I discuss a dynamical calculation that retains some of the von Neumann-Bocchieri-Loinger spirit but looks for statistical behavior over finite intervals of time.
and it is easy to show,” from Birkhoff s ergodic theorem,12 that Ps is always greater than or equal to the statistical expectation
Ps 2 Il(S) (3.3) Equality holds for ail sets S only if the classical motion is ergodic, so the survival probability gives us a variational criterion for ergodicity in classical mechanics. There is an interpretation of the classical survival probability that makes inequality (3.3) almost obvious: Ps is the fraction of time that a typical trajectory, starting in S, spends in S . If motion is ergodic and a typical trajectory from S wanders all over the energy surface, we expect it to spend less time “at home”, in S, than if motion is nonergodic and each trajectory from S is confined to a neighborhood of “home” that is smaller than the entire surface. In human terms, it’s Henry Kissinger, wandering the globe, compared with Archie Bunker, stuck in Queens; whom do you expect to find more at home? We shall be concerned with the finite-time behavior of the quantum survival probability, but it is instructive to begin by computing the infinite-time average. Suppose $(t) belongs to the N-dimensional energy shell spanned by eigenfunctions 41,..., d,,,, with eigenvalues E l , ..., EN, and assume for simplicity that the N eigenvalues are all different; then n
T
P = lim TI& dt P(t) T--
=
Clan14 n
(3.4c)
As expected, the infinite-time average P depends only on the expansion coefficients, not on the energy spectrum; what is worse, P should equal 1 / N for statistical behavior, but n
and therefore
which is greater than zero unless Jcxnl* = 1 / N for all n. The classical variational criterion, applied literally in quantum mechanics, would say that essentially all quantum motion is nonergodic, no matter what the spectrum of the Hamiltonian.
(11) P. Pechukas, Chem. Phys. Lett., 86, 553 (1982). (12) G. D. Birkhoff, Proc. Natl. Acad. Sci. U.S.A. 17, 656 (1931).
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984
4826
Now for finite-time P ( t ) calculations; many have been reported in the past few years,13 and I think it is fair to describe this literature as contentious. An individual P ( t ) curve depends on both the choice of initial state and the choice of Hamiltonian, and one can argue about either. My contribution to this literature” can be described as a coward’s P(t) calculation: to avoid argument about the choice of initial state, I average over all initial states, as in section 11; to avoid argument about the choice of Hamiltonian, I average over all Hamiltonians. P ( t ) (eq 3.4b) depends on differences of energy eigenvalues-that is, on spacings between energy levels- so the average over Hamiltonians amounts to an average over level spacings, and the question is, which level spacing distribution is appropriate for the quasi-periodic, “regular” regime, and which for the chaotic, “irregular” regime? The level spacing distribution in the regular regime was deduced by Berry and Tabor,14 using the formulas of semiclassical quantization appropriate to this regime. It is random: the probability distribution of the spacing between adjacent levels is Poisson, p ( S ) = exp(-S) where S = AE/ ( A E ) and ( A E ) is the mean level spacing; successive level spacings are independent. The doubly averaged P(t)-averaged over initial state and Hamiltonian-is given by
Pechukas 1.c
0.8
0.6
: 0.4
0.2
0
I
1
I
I
2
3
io*’
t / h
(3.7)
Figure 1. ( ( P ( r ) ) )vs. ( A E ) t / h for the case N = 5 : (---)
regular
regime; (-) irregular regime.
and m#n
C (exp[i(E, - E n ) t / h l ) = m>n x ~ =-2 ~ Re [xN+l - Nx2 + ( N - I)x] /( 1 - x ) ~
2 Re 2 Re m>n
(3.8) where
x = (exp(iAEt/h)) = (1 - i(AE)t/h)-l
(3.9)
The distribution of energy eigenvalues to be expected in the irregular regime will be discussed in the next section; the probability distribution p(S) for the spacing between adjacent levels is peaked at nonzero S, in contrast to the Poisson distribution, and successive level spacings are strongly correlated. In a word, the spectrum is ordered, not random. I cannot evaluate the average ( ( P ( t ) ) )using this eigenvalue distribution. Instead, as a model for the irregular spectrum, let us take perfect order-absolutely equal level spacings, p ( S ) = 6(S - 1). Then eq 3.7-3.8 still hold, but with x = exp(i(AE)t/h). It may seem insane, to take as the model for quantum chaos a harmonic oscillator spectrum of equally spaced levels, but one should keep in mind that this is in fact the spectral pattern in any narrow energy range for a familiar class of ergodic systems, one-dimensional vibrations in a simple potential well; and although the irregular spectrum in systems with more than one degree of freedom may not be perfectly ordered, the paradox remains: order in the dynamics is associated with disorder in the energy spectrum, and vice versa. Notice from eq 3.7 that in both the regular and irregular regimes the infinite-time average of ( ( P ( t ) ) )is 2/(N l), which is almost twice statistical. It is in the finite-time behavior that we see the difference between the regimes. In each of the two cases, ( ( P ( t ) )) depends only on N and on the dimensionless time ( A E ) t / h .There are two characteristic times in the problem, the short time determined by N ( A E ) t E a h , which is the time for the initial decay of ( ( P ( t ) ) )due to dephasing across the energy shell, and the long time determined by (AE)t N ah after which
+
(13) P. Brumer and M. Shapiro, Chem. Phys. Lett., 72, 528 (1980); K. G. Kay, J . Chem. Phys., 72, 5955 (1980); M. J. Davis, E. B. Stechel, and E. J. Heller, Chem. Phys. Lett., 76, 21 (1980); Y. Weissman and J. Jortner, ibid., 78, 224 (1981); Y. Weissman and J. Jortner, Phys. Lett. A , 83, 55 (1981); J. S. Hutchinson and R. E. Wyatt, Phys. Rev. A, 23, 1567 (1981). (14) M. V. Berry and M. Tabor, Proc. R.Soc. London, Ser. A, 356,315 (1977).
the quantum dynamics become sensitive to individual pairs of energy levels. It is the period between these characteristic times that is of physical interest. Figure 1 shows ( ( P ( t ) ) )for the case N = 5 from t = 0 to t = ?rh/ ( AE). In the regular regime ( (P(t ) ) ) decays monotonically to its long-time average 2/(N + 1); the motion is nonstatistical. In the irregular regime, after the initial rapid dephasing, ( ( P ( t ) ) ) fluctuates around the statistical value 1/N; for example, at t = ?rh/(AE) weseefromeq3.7-3.10 that ( ( P ( t ) ) ) = I / N i f N i s odd, 1 / ( N 1) if N is even. For large N, ( ( P ( t ) ) )is almost always close to the statistical value 1/N. A “typical” system, in the irregular regime, “forgets” its initial state after the initial dephasingLsand is no more likely to be in that initial state than in any other; in the regular regime a “typical” system, even after dephasing, is roughly twice as likely to be in its initial state. This is a striking result. Is it also significant? I do not know. The calculation confirms that classically ergodic one-dimensional vibrations ought also to be regarded as ergodic in quantum mechanics, but that is not major news. It would be very helpful to have ( ( P ( t ) ) )calculated with the generic joint eigenvalue distribution for the irregular regime, eq 4.4, in order to know whether the spectral pattern characteristic of “quantum chaos” is in fact associated with appropriately “statistical” quantum dynamics; but I cannot do the calculation.
+
IV. Distribution of Energy Eigenvalues in the Irregular Spectrum It is often useful, and in this section it will be essential, to regard Planck‘s constant h as a variable parameter. Indeed, the whole question of “quantum chaos” makes most sense in the semiclassical limit h 0; certainly in the other direction, for large h , it is nonsense. For example, consider bounded quantum motion in a potential that supports only a finite number of bound states. If h is made large enough, the potential will support only a single bound state; is there then any point in asking whether the system, sitting forever in its single stationary state, is chaotic or not? The message of this section is that two common prejudices about 0 are in fact the nature of “quantum chaos” in the limit h related.
-
-+
(15) Except at intervals o f f = 2 r h / ( A E ) , when the initial state recurs and ( ( P ( t ) ) )rises briefly to one. These ‘blips” in ( ( P ( t ) ) )contribute ca. 1/N to the infinite-time average, which is why ( ( P ( t ) ) )in the irregular regime can be statistical almost all the time yet have the nonstatistical infinite-time average of 2 / ( N 1).
+
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4821
Remarks on Quantum Chaos The first prejudice concerns the spatial pattern of eigenfunctions; it is that eigenfunctions of roughly the same energy look roughly the same: each is spread over the entire classically allowed region of configuration space appropriate to its energy, with coarsegrained probability density that agrees well at each point in space with the classical microcanonical density a t that energy.16 The 0 reasoning behind this hypothesis is that in the limit h quantum states must be associated with dynamically invariant phase space manifolds, and for a chaotic system there is only one such, the entire energy surface. The second prejudice concerns the spectral pattern of eigenvalues; it is that the eigenvalues in the irregular spectrum should be distributed as are those of a random matrix.” The reasoning behind this hypothesis is that in the irregular regime there are no constants of the motion other than the Hamiltonian, and thus no a priori restrictions on the matrix elements of the Hamiltonian with respect to an arbitrary basis. There is some support from numerical calculations for both the “irregular eigenfunction” hypothesis and the “random matrix” hypothesis,’* but the semiclassical theory of eigenfunctions and eigenvalues in the irregular regime is not nearly in so satisfactory a state as that for the regular regime. And the classical-quantum correspondence will never be perfect in the irregular regime; for one thing there are classically ergodic systems that cannot be ergodic quantum mechanically, because of symmetry-enforced degeneracies in the quantum spectrum.lg Such systems have an extra quantum constant of the motion which has no classical counterpart. In this section, then, the term “irregular” will be reserved for quantum systems having “irregular” eigenfunctions, without reference to the classical mechanics of such systems. The distribution of energy eigenvalues in the irregular spectrum will be deduced from the spatial distribution of the corresponding eigenfunctions: the “irregular eigenfunction” hypothesis implies the “random matrix” hypothesis.20 Berry and Tabor14 studied the regular spectrum in the limit h -,0 by use of the formulas of semiclassical quantization. There are no such formulas for the irregular spectrum, so a different approach is necessary; I use the “equations of motion” governing the eigenvalues as h varies. Suppose the Hamiltonian is H = -h2D V where D is a real differential operator in d > 1 dimensions, Vis the potential, and the spectrum of H is discrete. Let h2 = exp(-A), so that h -,0 is X m, and let {$,(A)) be orthonormal real eigenfunctions of H , with energies (E,,(X)]. The spectrum of H may be assumed to be nondegenerate; accidental degeneracies will be broken as X varies, symmetry-enforced degeneracies imply that the system cannot be considered ergodic. Then the “equations of motion” for E,@) and the various matrix are elements V,,(X) = (C$,(X),V+,(X)) dE,/dX = V,,, - E, (4.la) -+
+
-
dV,,/dX = 2 C V,,Z/(E, - E,)
(4.lb)
mfn
dV,,/dX
= C V,/V/,{(E, /#m,n
- E/)-’
+ (E, - E/)-’) +
VrnAVrnrn - J’nn)/(En - E m ) ( 4 . 1 ~ ) To see that the spatial pattern of the eigenfunctions can determine the spectral pattern of the eigenvalues, consider first the regular regime. “Regular” eigenfunctions of roughly the same energy typically look very different, because that energy is divided (16) K. S.J. Nordholm and S. A. Rice, J. Chem. Phys., 61,203 (1974); M. V. Berry, Phil. Trans. R. SOC.London, Ser. A , 287, 237 (1977);M.V. Berry, J . Phys. A , 10, 2083 (1977);A. Voros, Lecture Notes Phys., 93,326 (1979). (17) For random matrix theory, see the very useful reprint collection by C. E. Porter, ”Statistical Theories of Spectra: Fluctuations”,Academic Press, New York, 1965. (1 8) J. S.Hutchinson and R. E. Wyatt, Chem. Phys. Lett., 72,378 (1980); S . W. McDonald and A. N. Kaufman, Phys. Reo. Lett., 42,1189 (1979); G. Casati, F. Valz-Gris, and I. Guarneri, Lezr. Nuovo Cimento, 28, 279 (1980); M. V. Berry, Ann. Phys., 131, 163 (1981);V. Buch, R. B. Gerber, and M. A. Ratner, J . Chem. Phys., 76,5397 (1982). (19) E.J. Heller, Chem. Phys. Lett., 60, 338 (1979); P. Pechukas, J. Chem. Phys., 78, 3999 (1983). (20) P. Pechukas, Phys. Rev. Lett., 51, 943 (1983).
in very different ways among the “vibrational modes” in potential V. For small h the wave functions are highly oscillatory, and the matrix element V,, of the smooth potential V is very smalltypically, exponentially small with h , V, = O(exp(-()/h)). The spacings between adjacent energy levels are O(hd),so for small h , in the regular regime, Vmatrix elements among neighboring states are negligible compared to the level spacings. Furthermore, the diagonal elements V, and V,, are typically very different, because states n and m , with very different spatial distributions, “sample” very different regions of the potential; in order of magnitude, V,, - V, is O(l), independent of h . Consider N consecutive levels; think of their energies as “positions” in one dimension, and X as the “time”. The ”velocities” of these “particles”, determined according to eq 4.la by the diagonal matrix elements of V, are all very different; the “particles”, coupled according to eq 4.lb by the off-diagonal matrix elements, are essentially noninteracting. For so long as we can also ignore interactions of the levels with their “distant” neighbors, the N energy levels move as a one-dimensional ideal gas of particles with various velocities. Each state couples strongly (V,, = O(1)) to states that are similar to it in the assignment of vibrational quantum numbers; these are O ( h ) away in energy; from eq 4.lb the “acceleration” due to these interactions is O(h-’) and can be ignored if we follow t > 0. During this “time” the levels for a “time” AX = O(hl+‘), each level makes vast numbers of collisions with its near neighbors, provided d > 1, because the “mean free path”-the energy level spacing-is O(hd). At any “instant”-that is, at any particular value of h-the energy levels distribute themselves along the energy axis as one-dimensional free particles distribute themselves along the line: at random, with a Poisson distribution of nearest-neighbor spacings. This argument by statistical mechanics is no substitute for the beautiful direct analysis of Berry and Tabor,14who not only obtain 0 when that disthe limiting energy level distribution as h tribution exists but also show that for certain potentials the limiting distribution does not exist; but it conveys the essence of the situation. In the irregular regime, the hypothesis that all eigenfunctions of roughly the same energy look roughly the same has two consequences for the eigenvalue equations of motion. First, there are no strong selection rules in the irregular regime.’ We cannot throw away any of the off-diagonal matrix elements of V, because they are all of the same order of magnitude in h. Second, the diagonal Vmatrix elements are roughly the same for states of roughly the same energy, so the energy eigenvalue curves E,@) run roughly parallel to each other. The distribution of eigenvalues in the irregular regime is determined by fluctuations about the mean drift of the energy levels. What are the orders of magnitude? As in the regular regime, a given state n couples mainly to states that lie within O ( h ) in energy from E,; states further away in energy differ so much from 4, in “local wavelength” that the V,, integral is negligible. The mean level spacing is O(hd), so state n couples to O(hl-“) others, and from the sum rule ~ , + , , V r n=~(&, (u‘ - V,,:)&) = 0(1) it follows that V,: = O(hhl). The fluctuation in diagonal matrix elements should be of the same order of magnitude, V,: - ( V,,,)* = O(hd-1). In the regular regime, with strong selection rules, V matrix elements between nearby states are small compared to the mean level spacing; in the irregular regime, with no selection rules, they are large. Consider a “collision” between adjacent levels E,(X) and En+l(X).Typically, in both the regular and irregular regimes, a “collision” results in an avoided rather than an actual crossing of the levels, for crossing requires that the V matrix element between the two states vanish at the same “instant” X as does their energy difference, and by the familiar argument for the noncrossing rule this is an accident with zero probability. In the regular regime the “duration” of the “collision”-the width AX of the avoided crossing-is exponentially small with h , for the levels will not interact appreciably until their separation is as small
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The Journal of Physical Chemistry, Vol. 88, No. 21, 1984
as the V matrix coupling between them. The duration of the co!lision is therefore negligible compared to the “time” O( hd) between collisions; in the regular regime, avoided crossings are isolated, essentially “instantaneous” events in which two levels simply exchange “velocities”. In the irregular regime, on the other hand, the “duration” of the collision is comparable to the “time” between collisions, as one can see from the following estimates: according to eq 4.la the “velocity of approach” of adjacent levels is determined by fluctuations in diagonal V matrix elements and is therefore O( h(“’)I2);from eq 4.1 b the “acceleration” is O(h“l)/O(hd) = O(h-’); the “time” needed to reverse the “velocity of approach”-the duration of the collision-is therefore O ( h ( d - 1 ) / 2 ) / O ( h=- l 0) ( h ( d + 1 ) / 2which ) , is comparable to the “time” between collisions, O( hd)/O(h(d-1)/2). Loosely put, as h varies every level in the irregular regime is always in a state of avoided crossing with its various neighbors.z1 Now follow N >> 1 consecutive levels, centered around Eo at ho,for a short interval AA centered on Xo, To remove the mean level drift, let e,(A) = E,(X) - E(A) where E(A) is determined by dE(A)/dX = ( V),- E(h), E(Ao)= Eo, and ( V ) , is the classical microcanonical average of V a t E(X): ( V), = l d r V 6(E(X) H ) / J d r 6(E(X) - H). Then v,,(A) = V,,(X) - (V), is the fluctuation in diagonal matrix element, and, with the notation u,, V,, for the off-diagonal elements, eq 4.1 still hold when all E‘s and Vs are replaced by e’s and u’s, with one exception: an extra term, -d( V),/dA, appears on the right side of eq 4.lb. However, we are about to kill that extra term, as part of the following three approximations in the system of equations: 1. On the right side of eq 4.la e,, which is O(hod)and therefore is dropped. A similar connegligible beside u,, = O(f~o(~-’)/*), sideration of orders of magnitude allows us to drop the extra term in eq 4.1 b mentioned above: it is O( l ) , while the sum we retain is O( h0-l). 2. Each level is allowed to interact only with its ”nearest neighbors; Le., u,, is set equal to zero for Irn - nl > N’where 1 CC N’CC N . The justification for this can be seen from eq 4.lb: because of the energy denominator, the average interaction of a given state with its distant neighbors a certain amount up in energy is almost balanced by the average interaction with states the same distance down in energy. 3. To get a finite system of equations, “periodic boundary conditions” are adopted; Le., we imagine N energy levels el < ... C arranged on a circle of length N ( AE)o,where (AE), is the mean level spacing at &, each level interacting with its ”nearest neighbors according to eq 4.1, under the convention that eN+j = ej N(AE),,. After these modifications eq 4.1 scale nicely with hoeLet r = X/hn(d+1)/2 and define new variables x,(T) and u,.(r), which are o(i)in magnitude, by e,(A) = x,(r)(AE)o, U,,(ij = ( T ) ( AE)O/ho(d+l)/z.Then
+
dx,/dr = u,,
(4.2a) (4.2b)
du,,/dr = Cum/u/n((xm- xl1-I
+ (xn - XI)-’) + urnn(umrn - unn)/(xn
- xm) (4.2~)
We are now dealing with N points x1 C ... C xNmoving on a circle of length N. For how long a ”time” r do we follow them? AX should be short, so that the fractional change in h is small; say, AA = O(hof), e > 0, so that AA 0 as ho 0. But the scaled interval A r = AA/hO(d+1)/2goes to infinity, so to find the distribution of energy levels in the irregular spectrum-to find the distribution of the N points {x,,)on a circle of length N-we must integrate eq 4.2 for an arbitrarily long “time” 7. We should therefore apply the methods of classical statistical mechanics. The “dynamical variables” are the positions {x,)and the various matrix elements {urn”).We want to find an “equilibrium” prob-
- -
~~~~~
(21) D. W. Noid, M. L. Koszykowski, and R. A. Marcus, Chem. Phys. Lett., 73, 269 (1980).
Pechukas ability density p in the space of these variables-that is, a timeindependent solution of the continuity equation dp/dr + div(pv) dp/dr + p div u = 0. From eq 4.2 div u =
Caa,,/au,, = C(u,, - u,,,,)/(x, - x,) = E(& - X,,)/(x,, - xm)= -(d/dT) In nix, - x,I (4.3)
where the sum and the product are over interacting pairs of levels, each pair counted once. The continuity equation can therefore be written d/dr(p/n(x, - x,l) = 0 and any p of the form cnlx, - xml, where c is a properly normalized constant of the motion, will be an “equilibrium” density. It is easy to check that tr(u) and tr(u2) are constants of the motion according to eq 4.2. Let us be optimistic and assume there are no others. Then any constant c is a function only of u-matrix elements and does not involve the {x,,),so the joint distribution of N eigenvalue positions x, on the circle of length N is xN)
nix, - XmI
(4.4)
which is the essential result of this section. There is a second way to “derive” eq 4.4 which is criminally sl~ppyas mathematics but perhaps more satisfying as physics than the argument above. The equations of motion for the off-diagonal elements (eq 4 . 2 ~ are ) annoying; ignore them-pretend that the off-diagonal u,, elements are constant in T . Then eq 4.2a-b are Hamilton’s equations of motion for N particles of unit mass under the repulsive potential U = -2Zu,,2 In Ix, - xml,where the sum is over interacting pairs. Assume canonical equilibrium at an “eigenvalue temperature” T, where T i s determined by the mean square fluctuation in diagonal matrix element according to the To find the distribution ”energy” equation (u,:) = k T 8’. of the x,’s, average exp(-@U) over the distribution of off-diagonal elements urn,. Be content with the first cumulant, so the distribution of xis is proportional to exp(2@C( urn:) In Ix, - x,l). Now each “irregular” eigenfunction is an essentially random linear combination of a large number of “regular” functions. Therefore u, should have the same statistics as the expectation value of a (large) matrix in a state selected at random, while u,, should have the same statistics as a transition matrix element of the same matrix between two orthogonal states selected at random. In a real linear vector space it is easy to calculate then that (u,:) = 2(umnZ),so 2@(um,2)= 1 and we are led directly to eq 4.4. What is the distribution of spacings between adjacent energy levels predicted by eq 4.4? In the limit N m, N ’ - 0 3 , it must be the same as in Dyson’s “circular ensemble”,22which differs from eq 4.4 only in that the distance between points on the circle is Cartesian distance in two dimensions rather than arc length around the circle. Dyson showedz3that his distribution in turn is equal to the large-matrix limiting distribution in the real Gaussian ensemble, which has been beautifully analyzed by Mehtaz4 and GaudinZ5and found to be very close-within 5% except in the tail of the distribution-to Wigner’s guess,26p ( S ) = (?r/2)S exp(-aS2/4). I like this derivation of the eigenvalue distribution in the irregular spectrum, but I confess to nervousness on one point. The equations of motion (4.2) themselves have equilibrium solutions, x, = n, u, = 0, and u,, any function of Im - nl;Le., equally spaced energy levels with interactions that depend only on the distance between levels. These are the solutions appropriate for one-dimensional vibrations. Are these solutions stable, and if so, how does that affect the statistical mechanics leading to eq 4.4? -+
V Postscript The uncharitable view of the study of “quantum chaos”, that it is a racket with a dismally low average in the solid science per (22) F. J. Dyson, J . Math. Phys., 3, 140 (1962). (23) F. J. Dyson, J. Math. Phys., 3, 166 (1962). (24) M. L. Mehta, Nucl. Phys., 18, 395 (1960). (25) M. Gaudin, Nucl. Phys., 25, 447 (1961). (26) E. P. Wigner, “Proceedings of the Conference on Neutron Physics by Time-of-Flight”, Gatlinsburg, Tennessee, 1956, ORNL-2309, 1957, p 59, reprinted in ref 17.
J . Phys. Chem. 1984,88, 4829-4839 printed page statistic, is not entirely without merit. The formal structure of quantum mechanics is much simpler than that of classical mechanics and does not support the rich variety of distinctions in dynamical behavior that one has with the latter. Even von Neumann had to strain to find chaos in the orderly abstract dynamics of quantum systems, and as we have seen it was not his most successful effort. I have tried to make two points in this paper: first, it is the correspondence limit h 0 in which one should look for solid evidence of statistical behavior in quantum systems; and second, it is finitetime behavior that is of real interest in quantum mechanics, not the infinite-time limit used in classical ergodic theory. Perhaps there is even a challenge here to classical
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theory: given only a finite segment of a classical trajectory, is there any meaningful measure of how ”chaotic” that trajectory is? Certainly the surfaces of section that a theoretical chemist uses to illustrate classical chaos are generated by finite calculation! Finally, it is encouraging that experimentalists are now interested in this subject, and wonderful that some of the predictions for the irregular spectrum of highly vibrationally excited molecules have now been confirmed by stimulated emission pumping experiments.*’ (27) E. Abramson, R. W. Field, D. Imre, K. K. Innes, and J. L. Kinsey, J . Chem. Phys., 80, 2298 (1984).
Classical Liouville Mechanics and Intramolecular Relaxation Dynamics Charles Jaff$ and Paul Brumer*t Department of Chemistry, University of Toronto, Toronto, Ontario, Canada M5S 1A1 (Received: February 21, 1984)
The formal Hilbert space structure of classical mechanics is reviewed with emphasis on the relationship of the spectrum of the Liouville operator to regular vs. irregular motion. Two aspects of this approach are then described. First, eigendistributions of the Liouville operator for the harmonic oscillator and pendulum are displayed, providing insight into this alternate view of classical mechanics. Second, the dynamics of selected classical distributions in regular systems are discussed, with emphasis on the broad range of possible behavior, including periodicity, dephasing, and relaxation.
I. Introduction Classical mechanics has, for several decades, been useful in studies of molecular dynamics.’ Interest in conditions for the applicability of classical mechanics, particularly in intramolecular dynamics,2 has grown in recent years as a consequence of the recognition that classical conservative Hamiltonian systems undergo a transition, with increasing energy, from regular quasiperiodic motion to irregular or chaotic dynamic^.^ Although this change in character justifies historical concepts of statistical relaxation in highly excited isolated molecules, firm connections with quantum mechanics have yet to be e~tablished.~ At least some of the difficulty which arises in analyzing relationships between classical and quantum dynamics stems from the distinctly different approaches advocated to study them. In the former case dynamics is usually described via Hamiltonian mechanics, with trajectories as the focus. In the latter case, however, the quantum uncertainty principle insists, a priori, on the study of distributions with inherent position-momentum widths. The purpose of this paper is (a) to advocate the distribution dynamics viewpoint in the purely classical framework and (b) to emphasize the conceptual utility of the Liouville eigenfunction formulation which provides an alternate picture of classical mechanics and a useful view of relaxation phenomena. The specific technical emphasis is on the nature of the spectrum of the Liouville operator and its relationship to regular vs. irregular motion, as well as on the nature of relaxation in classically regular systems. As such, our goal is distinct from previous sporadic effortsS to utilize the Liouville equation as a computational tool or formulations in statistical mechanics which focus on the thermodynamic limit.6 The basic approach of interest below is the Liouville formulation of classical mechanics in which a distribution p is defined on phase space (p,q) and evolves in time in accord with the Liouville equation: ‘Current address: Department of Chemistry, Columbia University, New York, New York 10027. *I. W. Killam Research Fellow (1981-1983).
0022-3654/84/2088-4829$01 .50/0
Here H is the (conservative) Hamiltonian, { , is tHz: Poisson bracket, and p,q are any set of generalized canonical momenta and coordinates. Trajectories, which arise by choosing p(p,q;O) = 6(p-po) 6(q-qo),are the characteristics of Liouville’s equation and, as such, provide an equivalent dynamics. However, fundamental arguments have repeatedly been advanced in favor of p(p,q;t)dynamics over trajectory dynamics. These include those due to the Ehrenfests7 and to Born* in which recognition is made of in-practice limitations of classical measurement devices which prevent the preparation of an initial phase space point. More recently, Brillouin9 and Prigogine” have formulated such arguments with explicit reference to the rapid growth of errors associated with the incomplete specification of the initial state in the regime of irregular motion. Prigogine, in partittdar, argues for (1) E.g., M. Karplus, R. N. Porter, and R. D. Sharma, J . Chem. Phys., 43, 3259 (1965); D. L. Bunker and M. Pattengill, ibid., 48, 772 (1968); L. Verlet, Phys. Reu., 165, 201 (1968). (2) See, e&: (a) Discuss. Faraday SOC.,75 (1983). (b) Adu. Chem. Phys., 47 (1981)-both containing several relevant articles. (3) For reviews, see: M. V. Berry in “Topics in Nonlinear Dynamics”, S. Jorna, Ed., American Institute of Physics, New York, 1978; P. Brumer, Adu. Chem. Phys., 47, 201 (1981); S . A. Rice, ibid., 47, 117 (1981). (4) E&: (a) D. W. Noid, M. L. Koszykowski, and R. A. Marcub, Ann. Rev. Phys. Chem., 32, 267 (1981); (b) K. Kay, J . Chem. Phys., 72, 5955 (1980). . ( 5 ) W. H. Miller and B. M. Skuse, J. Chem. Phys., 68, 295 (1978); B. C. Eu, ibid., 54, 559 (1971) are but a few examples. (6) I. Prigogine, “Non-Equilibrium Statistical Mechanics”, Wiley, New York, 1962; R. S. Zwanzig in “Lectures in Theoretical Physics”, Vol. 3, Wiley, New York, 1961, p 135. (7) P. Ehrenfest and T. Ehrenfest, “Encyklopadia der Matimatischen Wissenschaften IV”, 32, Leipzig, 1911. ( 8 ) M. Born and D. J. Hooten, Proc. Cambridge Philos. SOC.,52, 287 (1956), and references therein. (9) L. Brillouin, “Scientific Uncertainty and Information”, Academic Press, New York, 1d64. (10) I. Prigogine, “From Being to Becoming”, W. H. Freeman, San Francisco, 1980.
0 1984 American Chemical Society
